1. Completely Randomized Design9 17

2. Completely Randomized Design
1. Experimental Units (Subjects) Are Assigned Randomly to Treatments
Subjects are Assumed Homogeneous
2. One Factor or Independent Variable
2 or More Treatment Levels or Classifications
3. Analyzed by One-Way ANOVA

3. Randomized Design Example
Are the mean training times the same for 3 different methods?
9 subjects
3 methods (factor levels)


 78

4. yij = the observation in ith treatment and the jth replication
The Linier Model
i = 1,2,…, t
j = 1,2,…, r
yij = the observation in ith treatment and the jth replication
m = overall mean
t i = the effect of the ith treatment
eij = random error

5. One-Way ANOVA F-Test 9 17

6. One-Way ANOVA F-Test
1. Tests the Equality of 2 or More (p) Population Means
2. Variables
One Nominal Scaled Independent Variable
2 or More (p) Treatment Levels or Classifications
One Interval or Ratio Scaled Dependent Variable
3. Used to Analyze Completely Randomized Experimental Designs
Note:
There is one dependent variable in the ANOVA model. MANOVA has more than one dependent variable.
Ask, what are nominal & interval scales?

7. One-Way ANOVA F-Test Assumptions
1. Randomness & Independence of Errors
Independent Random Samples are Drawn for each condition
2. Normality
Populations (for each condition) are Normally Distributed
3. Homogeneity of Variance
Populations (for each condition) have Equal Variances

8. One-Way ANOVA F-Test Hypotheses
H0: 1 = 2 = 3 = ... = t
All Population Means are Equal
No Treatment Effect
Ha: Not All i Are Equal
At Least 1 Pop. Mean is Different
Treatment Effect
NOT 1  2  ...  t

9. One-Way ANOVA F-Test Hypotheses
H0: 1 = 2 = 3 = ... = t
All Population Means are Equal
No Treatment Effect
Ha: Not All i Are Equal
At Least 1 Pop. Mean is Different
Treatment Effect
NOT 1  2  ...  t
f(X) X  =  =  1 2 3
f(X) X  =   1 2 3

10. Why Variances? Observe one sample from each treatment group
Their means may be slightly different
How different is enough to conclude population means are different?
Depends on variability within each population
Higher variance in population  higher variance in means
Statistical tests are conducted by comparing variability between means to variability within each sample
In all cases, the population means are DIFFERENT! Should reject Ho.
Panel A:
Same treatment effect - means are equal.
Different random variation - standard deviations are different.
As variances WITHIN get bigger, we are more likely to conclude equal means.
In Populations 4, 5, & 6, it is possible to draw a sample and falsely conclude population means are equal.
Panel B:
Different treatment effect - means are different.
Same random variation - standard deviations are equal.
As variances AMONG get bigger, we are more likely to conclude population means are different.
In Populations 4,5, & 6, it is possible to draw a sample and falsely conclude population means are equal. 82

11. Two Possible Experiment Outcomes
Same treatment variation
Different random variation
Reject equality of means! A
In all cases, the population means are DIFFERENT! Should reject Ho.
Panel A:
Same treatment effect - means are equal.
Different random variation - standard deviations are different.
As variances WITHIN get bigger, we are more likely to conclude equal means.
In Populations 4, 5, & 6, it is possible to draw a sample and falsely conclude population means are equal.
Panel B:
Different treatment effect - means are different.
Same random variation - standard deviations are equal.
As variances AMONG get bigger, we are more likely to conclude population means are different.
In Populations 4,5, & 6, it is possible to draw a sample and falsely conclude population means are equal.
Can’t reject equality of means! 82

12. Two More Possible Experiment Outcomes
Same treatment variation
Different random variation
Different treatment variation
Same random variation A B
Reject
Reject
In all cases, the population means are DIFFERENT! Should reject Ho.
Panel A:
Same treatment effect - means are equal.
Different random variation - standard deviations are different.
As variances WITHIN get bigger, we are more likely to conclude equal means.
In Populations 4, 5, & 6, it is possible to draw a sample and falsely conclude population means are equal.
Panel B:
Different treatment effect - means are different.
Same random variation - standard deviations are equal.
As variances AMONG get bigger, we are more likely to conclude population means are different.
In Populations 4,5, & 6, it is possible to draw a sample and falsely conclude population means are equal.
Can’t reject equality of means! 82

13. One-Way ANOVA Basic Idea
1. Compares 2 Types of Variation to Test Equality of Means
2. Comparison Basis Is Ratio of Variances
3. If Treatment Variation Is Significantly Greater Than Random Variation then Means Are Not Equal
4. Variation Measures Are Obtained by ‘Partitioning’ Total Variation

14. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups. 84

15. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups. 85

16. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups.
Variation due to treatment 86

17. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups.
Variation due to treatment
Variation due to random sampling 87

18. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups.
Variation due to treatment
Variation due to random sampling
Sum of Squares Among
Sum of Squares Between
Sum of Squares Treatment
Among Groups Variation 88

19. One-Way ANOVA Partitions Total Variation
Variation due to Random Sampling are due to Individual Differences Within Groups.
Variation due to treatment
Variation due to random sampling
Sum of Squares Among
Sum of Squares Between
Sum of Squares Treatment (SST)
Among Groups Variation
Sum of Squares Within
Sum of Squares Error (SSE)
Within Groups Variation 89

20. Total Variation
Response, X X
Group 1
Group 2
Group 3

21. Treatment Variation
Response, X
X3 X
X2
X1
Group 1
Group 2
Group 3

22. Random (Error) Variation
Response, X
X3
X2
X1
Group 1
Group 2
Group 3

23. SS=SSE+SST

24. But

25. Thus, SS=SSE+SST

26. One-Way ANOVA F-Test Test Statistic
F = MST / MSE
MST Is Mean Square for Treatment
MSE Is Mean Square for Error
2. Degrees of Freedom
1 = t -1
2 = tr - t
t = # Populations, Groups, or Levels
tr = Total Sample Size

27. One-Way ANOVA Summary Table
Source of
Degrees
Sum of
Mean F
Variation of
Squares
Square
Freedom
(Variance)
n = sum of sample sizes of all populations.
c = number of factor levels
All values are positive. Why? (squared terms)
Degrees of Freedom & Sum of Squares are additive; Mean Square is NOT.
Treatment
t - 1
SST
MST =
MST
SST/(t - 1)
MSE
Error
tr - t
SSE
MSE =
SSE/(tr - t)
Total
tr - 1
SS(Total) =
SST+SSE

28. ANOVA Table for a Completely Randomized Design
Source of Sum of Degrees of Mean
Variation Squares Freedom Squares F
Treatments SST t SST/t-1 MST/MSE
Error SSE tr - t SSE/tr-t
Total SSTot tr - 1

29. The F distribution  F  Two parameters
increasing either one decreases F-alpha (except for v2<3)
I.e., the distribution gets smashed to the left  F F  ( v1 , v2 )

30. One-Way ANOVA F-Test Critical Value
If means are equal, F = MST / MSE  1. Only reject large F!
Reject H 
Do Not
Reject H F F a ( t  1 , tr
-t)
Always One-Tail!
© T/Maker Co.

31. Example: Home Products, Inc.
Completely Randomized Design
Home Products, Inc. is considering marketing a long-lasting car wax. Three different waxes (Type 1, Type 2, and Type 3) have been developed.
In order to test the durability of these waxes, 5 new cars were waxed with Type 1, 5 with Type 2, and 5 with Type 3. Each car was then repeatedly run through an automatic carwash until the wax coating showed signs of deterioration. The number of times each car went through the carwash is shown on the next slide.
Home Products, Inc. must decide which wax to market. Are the three waxes equally effective?

32. Example: Home Products, Inc.
Wax Wax Wax
Observation Type Type Type 3
Sample Mean
Sample Variance

33. Example: Home Products, Inc.
Hypotheses
H0: 1=2=3
Ha: Not all the means are equal
where:
1 = mean number of washes for Type 1 wax
2 = mean number of washes for Type 2 wax
3 = mean number of washes for Type 3 wax

34. Example: Home Products, Inc.
Mean Square Between Treatments
Since the sample sizes are all equal:
μ= (x1 + x2 + x3)/3 = ( )/3 = 29.8
SSTR= 5(29–29.8)2+ 5(30.4–29.8)2+ 5(30–29.8)2= 5.2
MSTR = 5.2/(3 - 1) = 2.6
Mean Square Error
SSE = 4(2.5) + 4(3.3) + 4(2.5) = 33.2
MSE = 33.2/(15 - 3) = 2.77 _ _ _ =

35. Example: Home Products, Inc.
Rejection Rule
Using test statistic: Reject H0 if F > 3.89
Using p-value: Reject H0 if p-value < .05
where F.05 = 3.89 is based on an F distribution with 2 numerator degrees of freedom and 12 denominator degrees of freedom

36. Example: Home Products, Inc.
Test Statistic
F = MST/MSE = 2.6/2.77 = .939
Conclusion
Since F = .939 < F.05 = 3.89, we cannot reject H0. There is insufficient evidence to conclude that the mean number of washes for the three wax types are not all the same.

37. Example: Home Products, Inc.
ANOVA Table
Source of Sum of Degrees of Mean
Variation Squares Freedom Squares F
Treatments
Error
Total

38. Using Excel’s ANOVA: Single Factor Tool
Value Worksheet (top portion)

39. Using Excel’s ANOVA: Single Factor Tool
Value Worksheet (bottom portion)

40. Using Excel’s ANOVA: Single Factor Tool
Conclusion Using the p-Value
The value worksheet shows a p-value of .418
The rejection rule is “Reject H0 if p-value < .05”
Because .418 > .05, we cannot reject H0. There is insufficient evidence to conclude that the mean number of washes for the three wax types are not all the same.

41. (Randomized Complete Block Design)
RCBD
(Randomized Complete Block Design)

42. Randomized Complete Block Design
An experimental design in which there is one independent variable, and a second variable known as a blocking variable, that is used to control for confounding or concomitant variables.
It is used when the experimental unit or material are heterogeneous
There is a way to block the experimental units or materials to keep the variability among within a block as small as possible and to maximize differences among block
The block (group) should consists units or materials which are as uniform as possible

43. Randomized Complete Block Design
Confounding or concomitant variable are not being controlled by the analyst but can have an effect on the outcome of the treatment being studied
Blocking variable is a variable that the analyst wants to control but is not the treatment variable of interest.
Repeated measures design is a randomized block design in which each block level is an individual item or person, and that person or item is measured across all treatments.

44. The Blocking Principle
Blocking is a technique for dealing with nuisance factors
A nuisance factor is a factor that probably has some effect on the response, but it is of no interest to the experimenter…however, the variability it transmits to the response needs to be minimized
Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units
Many industrial experiments involve blocking (or should)
Failure to block is a common flaw in designing an experiment (consequences?)

45. The Blocking Principle
If the nuisance variable is known and controllable, we use blocking
If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance (see Chapter 14) to statistically remove the effect of the nuisance factor from the analysis
If the nuisance factor is unknown and uncontrollable (a “lurking” variable), we hope that randomization balances out its impact across the experiment
Sometimes several sources of variability are combined in a block, so the block becomes an aggregate variable

46. Partitioning the Total Sum of Squares in the Randomized Block Design
SStotal
(total sum of squares)
SSE
(error sum of squares)
SST
(treatment
sum of squares)
SSB
(sum of squares
blocks)
SSE’
(sum of squares
error)

47. A Randomized Block Design
Individual
observations .
Single Independent Variable
Blocking
Variable 30

48. yij = the observation in ith treatment in the jth block
The Linier Model
i = 1,2,…, t
j = 1,2,…,r
yij = the observation in ith treatment in the jth block
m = overall mean
ti = the effect of the ith treatment
No interaction between blocks and treatments
rj = the effect of the jth block
eij = random error

49. Extension of the ANOVA to the RCBD
ANOVA partitioning of total variability:

50. Extension of the ANOVA to the RCBD
The degrees of freedom for the sums of squares in
are as follows:
Ratios of sums of squares to their degrees of freedom result in mean squares, and
The ratio of the mean square for treatments to the error mean square is an F statistic  used to test the hypothesis of equal treatment means

51. ANOVA Procedure
The ANOVA procedure for the randomized block design requires us to partition the sum of squares total (SST) into three groups: sum of squares due to treatments, sum of squares due to blocks, and sum of squares due to error.
The formula for this partitioning is
SSTot = SST + SSB + SSE
The total degrees of freedom, nT - 1, are partitioned such that k - 1 degrees of freedom go to treatments,
b - 1 go to blocks, and (k - 1)(b - 1) go to the error term.

52. ANOVA Table for a Randomized Block Design
Source of Sum of Degrees of Mean
Variation Squares Freedom Squares F
Treatments SST t – SST/t MST/MSE
Blocks SSB r - 1
Error SSE (t - 1)(r - 1) SSE/(t-1)(r-1)
Total SSTot tr - 1

53. Example: Eastern Oil Co.
Randomized Block Design
Eastern Oil has developed three new blends of gasoline and must decide which blend or blends to produce and distribute. A study of the miles per gallon ratings of the three blends is being conducted to determine if the mean ratings are the same for the three blends.
Five automobiles have been tested using each of the three gasoline blends and the miles per gallon ratings are shown on the next slide.

54. Example: Eastern Oil Co.
Automobile Type of Gasoline (Treatment) Blocks
(Block) Blend X Blend Y Blend Z Means
Treatment
Means

55. Example: Eastern Oil Co.
Mean Square Due to Treatments
The overall sample mean is 29. Thus,
SST= 5[( )2+ ( )2+ ( )2]= 5.2
MST = 5.2/(3 - 1) = 2.6
Mean Square Due to Blocks
SSB = 3[( ) ( )2] = 51.33
MSB = 51.33/(5 - 1) = 12.8
Mean Square Due to Error
SSE = = 5.47
MSE = 5.47/[(3 - 1)(5 - 1)] = .68

56. Example: Eastern Oil Co.
Rejection Rule
Using test statistic: Reject H0 if F > 4.46
Using p-value: Reject H0 if p-value < .05
Assuming  = .05, F.05 = 4.46 (2 d.f. numerator and 8 d.f. denominator)

57. Example: Eastern Oil Co.
Test Statistic
F = MST/MSE = 2.6/.68 = 3.82
Conclusion
Since 3.82 < 4.46, we cannot reject H0. There is not sufficient evidence to conclude that the miles per gallon ratings differ for the three gasoline blends.

58. Using Excel’s Anova: Two-Factor Without Replication Tool
Step 1 Select the Tools pull-down menu
Step 2 Choose the Data Analysis option
Step 3 Choose Anova: Two Factor Without Replication from the list of Analysis Tools
… continued

59. Using Excel’s Anova: Two-Factor Without Replication Tool
Step 4 When the Anova: Two Factor Without
Replication dialog box appears:
Enter A1:D6 in the Input Range box
Select Labels
Enter .05 in the Alpha box
Select Output Range
Enter A8 (your choice) in the Output Range box
Click OK

60. Using Excel’s Anova: Two-Factor Without Replication Tool
Value Worksheet (top portion)

61. Using Excel’s Anova: Two-Factor Without Replication Tool
Value Worksheet (middle portion)

62. Using Excel’s Anova: Two-Factor Without Replication Tool
Value Worksheet (bottom portion)

63. Using Excel’s Anova: Two-Factor Without Replication Tool
Conclusion Using the p-Value
The value worksheet shows that the p-value is
The rejection rule is “Reject H0 if p-value < .05”
Thus, we cannot reject H0 because the p-value = > a = .05
There is not sufficient evidence to conclude that the miles per gallon ratings differ for the three gasoline blends

64. Similarities and differences between CRD and RCBD: Procedures
RCBD: Every level of “treatment” encountered by each experimental unit; CRD: Just one level each
Descriptive statistics and graphical display: the same as CRD
Model adequacy checking procedure: the same except: specifically, NO Block x Treatment Interaction
ANOVA: Inclusion of the Block effect; dferror change from t(r – 1) to (t – 1)(r – 1)

65. Latin Square Design

66. Definition
A Latin square is a square array of objects (letters A, B, C, …) such that each object appears once and only once in each row and each column.
Example - 4 x 4 Latin Square.
A B C D
B C D A
C D A B
D A B C

67. The Latin Square Design
This design is used to simultaneously control (or eliminate) two sources of nuisance variability
It is called “Latin” because we usually specify the treatment by the Latin letters
“Square” because it always has the same number of levels (t) for the row and column nuisance factors
A significant assumption is that the three factors (treatments and two nuisance factors) do not interact
More restrictive than the RCBD
Each treatment appears once and only once in each row and column
If you can block on two (perpendicular) sources of variation (rows x columns) you can reduce experimental error when compared to the RCBD A B C D

68. Advantages and Disadvantages
Allows the experimenter to control two sources of variation
Disadvantages:
Error degree of freedom (df) is small if there are only a few treatments
The experiment becomes very large if the number of treatments is large
The statistical analysis is complicated by missing plots and mis-assigned treatments
Another variation – use a smaller Latin Square (3x3 or 4x4), but repeat it. Analysis can then be combined across the two replicates.

69. Selected Latin Squares
Latin Square Designs
Selected Latin Squares
3 x 3 4 x 4
A B C A B C D A B C D A B C D A B C D
B C A B A D C B C D A B D A C B A D C
C A B C D B A C D A B C A D B C D A B
D C A B D A B C D C B A D C B A
5 x 5 6 x 6
A B C D E A B C D E F
B A E C D B F D C A E
C D A E B C D E F B A
D E B A C D A F E C B
E C D B A F E B A D C

70. In a Latin square You have three factors:
Treatments (t) (letters A, B, C, …)
Rows (t)
Columns (t)
The number of treatments = the number of rows = the number of columns = t.
The row-column treatments are represented by cells in a t x t array.
The treatments are assigned to row-column combinations using a Latin-square arrangement 

71. Example
A courier company is interested in deciding between five brands (D,P,F,C and R) of car for its next purchase of fleet cars.
The brands are all comparable in purchase price.
The company wants to carry out a study that will enable them to compare the brands with respect to operating costs.
For this purpose they select five drivers (Rows).
In addition the study will be carried out over a five week period (Columns = weeks).

72. Each week a driver is assigned to a car using randomization and a Latin Square Design.
The average cost per mile is recorded at the end of each week and is tabulated below:

73. tk = the effect of the ith treatment ri = the effect of the ith row
The Linier Model
i = 1,2,…, t
j = 1,2,…, t
k = 1,2,…, t
yij(k) = the observation in ith row and the jth column receiving the kth treatment
m = overall mean
tk = the effect of the ith treatment
No interaction between rows, columns and treatments
ri = the effect of the ith row
gj = the effect of the jth column
eij(k) = random error

74. A Latin Square experiment is assumed to be a three-factor experiment.
The factors are rows, columns and treatments.
It is assumed that there is no interaction between rows, columns and treatments.
The degrees of freedom for the interactions is used to estimate error.

75. The Anova Table for a Latin Square Experiment
Source
S.S.
d.f.
M.S. F
p-value
Treat
SST
t-1
MST
MST /MSE
Rows
SSRow
MSRow
MSRow /MSE
Cols
SSCol
MSCol
MSCol /MSE
Error
SSE
(t-1)(t-2)
MSE
Total
t2 - 1

76. The Anova Table for Example
Source
S.S.
d.f.
M.S. F
p-value
Week 4
16.06
0.0001
Driver
21.79
0.0000
Car
22.24
Error 12
Total 24

77. Example
In this Experiment the we are again interested in how weight gain (Y) in rats is affected by Source of protein (Beef, Cereal, and Pork) and by Level of Protein (High or Low).
There are a total of t = 3 X 2 = 6 treatment combinations of the two factors.
Beef -High Protein
Cereal-High Protein
Pork-High Protein
Beef -Low Protein
Cereal-Low Protein and
Pork-Low Protein

78. In this example we will consider using a Latin Square design
Six Initial Weight categories are identified for the test animals in addition to Six Appetite categories.
A test animal is then selected from each of the 6 X 6 = 36 combinations of Initial Weight and Appetite categories.
A Latin square is then used to assign the 6 diets to the 36 test animals in the study.

79. In the latin square the letter
A represents the high protein-cereal diet
B represents the high protein-pork diet
C represents the low protein-beef Diet
D represents the low protein-cereal diet
E represents the low protein-pork diet and
F represents the high protein-beef diet.

80. The weight gain after a fixed period is measured for each of the test animals and is tabulated below:

81. The Anova Table for Example
Source
S.S.
d.f.
M.S. F
p-value
Inwt 5
111.1
0.0000
App
138.03
Diet
263.06
Error 20
3.181
Total 35

82. Diet SS partioned into main effects for Source and Level of Protein
d.f.
M.S. F
p-value
Inwt 5
111.1
0.0000
App
138.03 2
99.22
Level 1
820.88 SL
147.99
Error 20
3.181
Total 35

83. Graeco-Latin Square Designs
Mutually orthogonal Squares

84. Definition
A Greaco-Latin square consists of two latin squares (one using the letters A, B, C, … the other using greek letters a, b, c, …) such that when the two latin square are supper imposed on each other the letters of one square appear once and only once with the letters of the other square. The two Latin squares are called mutually orthogonal.
Example: a 7 x 7 Greaco-Latin Square
Aa Be Cb Df Ec Fg Gd
Bb Cf Dc Eg Fd Ga Ae
Cc Dg Ed Fa Ge Ab Bf
Dd Ea Fe Gb Af Bc Cg
Ee Fb Gf Ac Bg Cd Da
Ff Gc Ag Bd Ca De Eb
Gg Ad Ba Ce Db Ef Fc

85. The Graeco-Latin Square Design
This design is used to simultaneously control (or eliminate) three sources of nuisance variability
It is called “Graeco-Latin” because we usually specify the third nuisance factor, represented by the Greek letters, orthogonal to the Latin letters
A significant assumption is that the four factors (treatments, nuisance factors) do not interact
If this assumption is violated, as with the Latin square design, it will not produce valid results
Graeco-Latin squares exist for all t ≥ 3 except t = 6

86. Note:
At most (t –1) t x t Latin squares L1, L2, …, Lt-1 such that any pair are mutually orthogonal.
It is possible that there exists a set of six 7 x 7 mutually orthogonal Latin squares L1, L2, L3, L4, L5, L6 .

87. The Greaco-Latin Square Design - An Example
A researcher is interested in determining the effect of two factors
the percentage of Lysine in the diet and
percentage of Protein in the diet
have on Milk Production in cows.
Previous similar experiments suggest that interaction between the two factors is negligible.

88. For this reason it is decided to use a Greaco-Latin square design to experimentally determine the two effects of the two factors (Lysine and Protein).
Seven levels of each factor is selected
0.0(A), 0.1(B), 0.2(C), 0.3(D), 0.4(E), 0.5(F), and 0.6(G)% for Lysine and
2(a), 4(b), 6(c), 8(d), 10(e), 12(f) and 14(g)% for Protein.
Seven animals (cows) are selected at random for the experiment which is to be carried out over seven three-month periods.

89. A Greaco-Latin Square is the used to assign the 7 X 7 combinations of levels of the two factors (Lysine and Protein) to a period and a cow. The data is tabulated on below:

90. The Linear Model j = 1,2,…, t i = 1,2,…, t k = 1,2,…, t l = 1,2,…, t
yij(kl) = the observation in ith row and the jth column receiving the kth Latin treatment and the lth Greek treatment

91. tk = the effect of the kth Latin treatment
m = overall mean
tk = the effect of the kth Latin treatment
ll = the effect of the lth Greek treatment
ri = the effect of the ith row
gj = the effect of the jth column
eij(k) = random error
No interaction between rows, columns, Latin treatments and Greek treatments

92. A Greaco-Latin Square experiment is assumed to be a four-factor experiment.
The factors are rows, columns, Latin treatments and Greek treatments.
It is assumed that there is no interaction between rows, columns, Latin treatments and Greek treatments.
The degrees of freedom for the interactions is used to estimate error.